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Trig Apps Classwork Day 1: Law of Cosines Warm-up: 1. What is x to the nearest hundredth?
22. What is the solution set of the equation 3x 34 24 0?2 2 [A] {-2, 6} [B] {-12, } [C] {- , 12} [D] {-6, 2}3 3
x− − =
23. One root of the equation 2x 15 0 is
3 5 [A] [B] -3 [C] [D] 32 2
x− − =
Law of Cosines: Used to find _________________________________________________________________
Side a is opposite _____________________________
Side b is opposite_____________________________
Side c is opposite_____________________________
Conventions:________________________________________________________________
________________________________________________________________
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Example 1:
In ∆ABC, side b = 12, side c = 20 and m∠A = 45º.
Find side a to the nearest integer.
Example 2: In a parallelogram, the adjacent sides measure 40cm and 22cm. If the larger angle of the parallelogram measures 116º, find the length of the larger diagonal, to the nearest integer Example 3: Find the largest angle, to the nearest tenth of a degree, of a triangle whose sides are 9, 12 and 18 .
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Trig Apps Homework Day 1 1. In ∆ABC, a = 2, b = 4, and m∠C = 60º. What is the exact value of c? (1) 2 7 (2) 2 (3) 2 3 (4) 4 7 2. In ∆ABC, a = 3, b = 8, and m∠C = 120º. Find the length of side c to the nearest integer 3. Two sides of a parallelogram measure 17.0 and 20.0 centimeters. The measure of one angle of the parallelogram is 50º. Find, to the nearest tenth of a center, the length of the shorter diagonals.
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Trig Apps Classwork Day 2: More Law of Cosines Warm-up: 1. What is the solution set of the equation x2 – 5x = 0? [A] {0, 5} [B] {0} [C] {0, -5} [D] {5} 2. What is the positive solution of the equation 4x2 – 36 = 0?
3. A solution of the equation 2
94x
= is
3[A] 3 [B] 12 [C] 6 [D] 2
Examples: 1. Three sides of a triangle measure 20m, 30m, and 40m. Find the largest angle of the triangle to the nearest minute. 2. In a rhombus with a side of 24, the longer diagonal is 36. Find to the nearest degree, the larger angle of the rhombus. 3. In a parallelogram whose sides are 8 and 10, the measure of one the angles is 85º, find the length of the shorter diagonal to the nearest integer.
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Trig Apps Homework Day 2 1. In ∆ABC, if a = 6, b = 5, and c = 8, then cos A equals
75 53 3 53(1) (2) (3) - (4) 80 80 80 60
2. In ∆ABC, if a = 10, b = 7, and c = 8, then the value of cos C is
64 13 23 17(1) (2) (3) (4) 9 112 32 28
3. In ∆ABC, a = 10, b = 12, and m∠C is 41º 30’. Find the length of side c to the nearest integer 4. Two straight roads, Elm Street and Pine Street,
intersect creating a 40º angle, as shown in the accompanying diagram. John’s house (J) is on Elm Street and is 3.2 miles from the point of intersection. Mary’s house (M) is on Pine Street and is 5.6 miles from the intersection. Find, to the nearest tenth of a mile, the direct distance between the two houses.
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Trig Apps Classwork Day 3: Law of Sines Warm-up: 1. Solve ∆ABC (means find all missing pieces), given that a = 10, and b = 15 and c = 21 2. Find the missing side of the given triangle. [A] 15.87 in [B] 22.83 in. [C] 19.21 in [D] 521.14in Law of Sines:______________________________________________________________________________ Note: 1. Never are ALL 3 used at the same time 2. Remember product of means = prod of extremes Example 1: In ∆ABC, side a = 8, m∠A = 30º, and m∠C = 55º. Find side c to the nearest tenth of an integer
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Example 2:
In ∆DEF, sin F = 15
, sin D = 25
, and side f = 24
Find the length of side d. Example 3: In ∆ABC, m∠A = 19º 10’, m∠B = 32º 40’, and AB = 46.7. Find to the nearest tenth of a meter¸ the length of AC. ***To find a missing angle, another step is required Example 4: In the diagram, a = 55, c = 20, and m∠A = 110º. Find the measure of ∠C to the nearest degree.
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Regents Practice 1. Given a triangle with a = 20, C = 37º, and B = 24º, what is the length of c? Round the answer to two decimal places. [A] 29.59 [B] 13.52 [C] 13.76 [D] 43.01 2. Given a triangle with a = 8, C = 33º and B = 44º, what is the length of c? Round the answer to two decimal places. [A] 11.22 [B] 4.47 [C] 10.2 [D] 6.27 3. Given a triangle with a = 7, C = 17º and B = 32º, what is the length of c? Round the answer to two decimal places. [A] 9.97 [B] 12.69 [C] 2.71 [D] 3.86
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Trig Apps Homework Day 3
1. In ∆ABC, b = 12, c = 8, and sin B = 12
. Find the value of sin C.
2. In ∆RST, r = 7, sin R = 0.28, and sin T = 0.44. Find t. 3. In ∆ABC, m∠A = 60º, m∠B = 45º, and b = 4. What is the length of side a?
6 16(1) 2 6 (2) 2 2 (3) (4) 2 3
4. In ∆ABC, m∠A = 16º 30’, m∠B = 28º 10’, and AB = 34.5m. Find, to the nearest tenth of a meter, the length of AC 5. The Vietnam Veteran’s Memorial in Washington, DC
consists of two walls of black, polished granite, each 246.75 feet long, which meet at an angle of 125.2º. If extended, the west wall would reach to the Lincoln Memorial, 900 feet away from the end of the wall and the east wall would reach to the Washington Monument 3,500ft away from the end of the wall. Find the distance between the Lincoln Memorial and the Washington Monument to the nearest foot.
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Trig Apps Classwork Day 4: Area Warm-up: In ∆ABC, m∠A = 33º, a = 12, and b = 15. What is m∠B to the nearest degree? [A] 41 [B] 43 [C] 48 [D] 44 Review: When looking for a side or an angle: If you have SAS or SSS, use Law of _______________, Anything else use Law of ___________________ Trig and Area: Basic formula for Area of a triangle:________________________
Trig Area of a triangle A∆ = 1 ba sin2
C OR A∆ =
**the letters of the formula may change, remember: _______________________________________________ Example 1 Given the triangle at the right, find its area. Express the area rounded to three decimal places.
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Practice with Area: Parallelograms Remember:
1) Draw parallelogram with __________________________ on bottom,
2) Draw ___________________________ in the bottom right (this is just like problem 5 on hw 4)
3) If looking for exact answer use reference triangles
4) Parallelograms are TWICE the area of one triangle or A = ab sinC
Example 2: Given the parallelogram at the right find its EXACT area Example 3: The accompanying diagram shows the floor plan for a kitchen. The owners plan to carpet all of the kitchen except the “work space” which is represented by scalene triangle ABC. Find the area of this work space to the nearest tenth of a square foot. Example 4: Gregory wants to build a garden in the shape of an isosceles triangle with one of the congruent sides equal to 12 yards. If the area of his garden will be 55 square yards, find, to the Nearest tenth of a degree, the three angles of the triangle
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Example 5: The accompanying diagram shows a triangular plot of land that is part of Fran’s garden. She needs to change the dimensions of this part of the garden, but she wants the area to remain the same. She increases the length of side AC to 22.5 feet. If ∠A remains the same, by how many feet should side AB be decreased to make the area of the new triangular plot of land the same as the current one? Example 6: The accompanying diagram shows the peak of a roof that is in the shape of an isosceles triangle. A base angle of the triangle is 50º and each side of the roof is 20.4 feet. Determine, to the nearest tenth of a square foot, the area of this triangular region.
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Trig Apps Homework Day 4
1. In ∆ABC, a = 6, b = 8, and sin C = 14
. Find the area of ∆ABC
2. In ∆ABC, which expression can be used to find the length of side a?
sin sin sin sin(1) (2) (3) (4) sin sin sin sin
B A b A b Bb A b B B A
3. In ∆ABC, m∠A = 30º, b = 14, and a = 10. Find sin B. 4. The lengths of the sides of ∆ABC are 5m, 7m and 8m a) Find, to the nearest ten minutes, the measure of the smallest angle of
the triangle. b) Using the result obtained in part a, find, to the nearest tenth of a
square meter, the area of ∆ABC 5. Two adjacent sides of a parallelogram measure of 10 and 12 cm. The angle included between these sides has a measure of 60º a) Find, to the nearest centimeter, the length of the shorter diagonal of
the parallelogram. b) What is the area of the parallelogram to the nearest square
centimeter?
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Trig Apps Classwork Day 5: Law of Sines – Ambiguous Case Warm-up: In ∆ABC, m∠A = 50º, m∠B = 35º, and a = 12. Find the missing sides and angle (nearest tenth, nearest degree) Ambiguous:____________________________________________________________________________ Facts to remember: 1. In a ∆, the sum of the interior angles is ________________
2. No triangle, can have ______________________________
3. The sine function has a range of _____________________
4. If the sin θ = pos. decimal < 1, then θ can lie in quad_____
(acute ∠), or in quad ________ (obtuse ∠)
Key Words: “how many distinct triangles” – ambiguous triangle problem 1) Draw triangle, split paper in half and list all angles on each side of the line 2) Use Law of Sines to find second angle. If angle is acute, subtract from 180º and place that ans. in the equivalent ∠ in the other ∆. 3) Find the third angle in each possible triangle 4) CHECK: the sum of the angles of each triangle. Are any rules broken? Example 1: In ∆ABC, a = 20, c = 16, and m∠A = 30º. How many distinct triangles can be drawn given these measurements?
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Example 2: In ∆ABC, a = 7, c = 16, and m∠A = 30º. How many distinct triangles can be drawn given these measurements? Example 3: In ∆ABC, a = 10 , c = 16, and m∠A = 30º. How many distinct triangles can be drawn given these measurements?
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Example 4: A landscape designer is designing a triangular garden with two sides that are 4 feet and 6 feet, respectively. The angle opposite the 4-foot side is 30º. How many distinct triangular gardens can the designer make using these measurements? Example 5 In ∆ABC, m∠A = 30, a = 14, and b = 20. Which type of angle is ∠B? [A] It must be an acute angle [B] It may be either an acute or an obtuse angle [C] It must be a right angle [D] It must be an obtuse angle Example 6: In ∆ABC, if AC = 12, BC = 11, and m∠A = 30, angle C could be [A] a right angle, only [B] an acute angle, only [C] either an obtuse angle or an acute angle [D] an obtuse angle, only
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Trig Apps Homework Day 5 1. Find the area of ∆ABC if a = 6, b = 12, and m∠C = 150º 2. In ∆ABC, m∠A = 50º, a = 48, b = 62, and ∠B is obtuse. Find m∠C to the nearest degree. 3. How many distinct triangles can be constructed if m∠A = 60º, a = 8, and b = 10? 4. If m∠A = 30º, side a = 8, and side b = 10, what is the total number of noncongruent triangles that can be constructed? 5. Using the data m∠A = 38º, b = 20, and a = 17, the number of distinct triangles that can be constructed is (1) one right triangle 2) two triangles 3) cannot be 90º 4) may be 90º 6. In rhombus, ABCD, m∠ABC = 100º 40’ and the length of each side is 5 in. a) What is the length of diagonal AC to the nearest tenth of an inch? b) Find the area of ABCD to the nearest square inch.
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Trig Apps Classwork Day 6: Forces/Vectors Warm-up: Sam is designing a triangular piece for a metal sculpture. He tells Martha that two of the sides of the piece are 40 inches and 15 inches, and the angle opposite the 40-inch side measures 120º. Martha decides to sketch the piece that Sam described. How many different triangles can she sketch that match Sam’s description? Parallelogram:___________________________________________________________________________ Properties (from 10R) 1. 2 sets of ____________________________ sides
2. 2 sets of ____________________________ sides
3. opposite angles ___________________________
4. consecutive angles ________________________
5. diagonals _______________________each other
Vectors & Forces A vector is a ________________________ force whose __________________ is represented by the
________________________ of a line segment and whose _______________________ is represented by the
____________________of the line segment.
When two forces act on the same object, the ____________________ is equal to the vector of the
__________________ of the parallelogram created by the vectors of the original forces.
Example
the red car is driving north at 60 miles per hour....and another car driving 72 miles per hour east slams into the red car... draw the picture.
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Vector Practice The accompanying diagram at the right shows a resultant force vector, R. Which diagram best represents the pair of component force vectors, A and B, that combined to produce the resultant force vector R?
Resolution of Force Practice **Draw picture first, place largest side on the bottom, and larger angle on the bottom right. 1. Two forces of 12 pounds and 20 pounds act on a body with an angle of 60º between them. Find the
magnitude of the resultant to the nearest pound. 2. If two forces of 30 pounds and 40 pounds act on a body with an angle of 150º between them, find the
magnitude of the resultant to the nearest pound.
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3. Two forces act on a body. The measure of the angle between the 34-pound force and the 40-pound resultant is 40º. Find the magnitude of the other force to the nearest pound.
4. Find to the nearest degree the measure of the angle between two applied forces of 8 and 10 lbs. if the
resultant is a force of 5 lbs. 5. Find to the nearest degree the measure of the angle between two applied forces of 18 and 10 lbs. if the
resultant is a force of 25 lbs. 6. Two forces act on a body so that the resultant is a force of 50 lbs. The measures of the angles between the
resultant and the forces are 25º and 38º. Find the magnitude of the larger applied force to the nearest lb.
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Trig Apps Homework Day 6
1. If, in ∆ABC, m∠A = 60º and m∠B = 30º, then ab
equals
1 1(1) (2) 2 (3) 3 (4) 2 3
2. In ∆ABC, m∠A = 30º, a = 6, and b = 10. Then ∠C (1) must be acute (2) must be obtuse (3) may be either acute or obtuse (4) may be a right angle 3. Sailboat S is 50 meters from a lighthouse located at point P. Fishing boat F is 65 meters from the same lighthouse. If m∠SPF is 102º 30’, find, to the nearest meter, the distance between the two boats 4. Two forces act on a body, making angles of 15º 40’ and 37º 30’ with the resultant. If the larger force is 42 pounds, what is the magnitude of the resultant to the nearest pound? 5. Find, to the nearest degree, the measure of the angle between two forces of 30lbs and 35 lbs, if the magnitude of the resultant is 42lbs. 6. Two forces act on an object. The first force has a magnitude of 80 pounds and makes an angle of 37º with the resultant. The magnitude of the resultant is 120 pounds. Find, to the nearest pound, the magnitude of the second force.
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Trig Apps Classwork Day 7: Double Triangles Warm-up: Point A is 100 ft on a horizontal line from point B. Point C, on a line with a 20º rise from the horizontal, is 80 ft from point B. Find the distance from A to C, to the nearest integer. Double Triangle Problems Given a triangle within a triangle and find one of the sides... Step 1: Draw the triangle and fill in all the information known
Step 2: Use the info and the law of sines to find a side of the smaller triangle (usually the hypotenuse)
Step 3: Use boring old Sin or Cos to find the missing info you need (NOT law of sines or law of cosines)
Examples 1. Dan lives in a tall apartment building. One day he heard a noise on the roof, so he went outside to
investigate. He walked a distance from his building, and turned to look up at the roof with an angle of elevation of 50º. He couldn’t see the roof well, so he walked 100 feet further, and turned to look at the roof with an angle of 20º. He was now able to see clearly that the antenna on top of the building had fallen over. How tall is Dan’s building?
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2. A fishing boat is approaching a lighthouse. The captain observes the angle of elevation of the light at the
top of the lighthouse to be 13º. After traveling 100 feet closer to the lighthouse, the angle of elevation of the light is now 20º. How tall is the lighthouse?
3. From a point A at the edge of a river, the measure of the angle of elevation of the top of a tree on the
opposite bank is 37º. From a point B that is 50 feet from the edge of the river and in line with point A and the foot of the tree, the measure of the angle of elevation of the top of the tree is 22º. Find, to the nearest foot, the width of the river.
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Trig Apps Homework Day 7 1. In ∆ABC, a = 20 and m∠C = 30º. For which value of b is the area of ∆ABC equal to 100 square units?
(1) 10 (2) 20 (3) 20 33
(4) 25
2. In ∆ABC, a = 5, b = 10, and sin A = 12
. Which is true of ∆ABC?
(1) It must be a right triangle (2) It must be an acute triangle (3) It must be an obtuse triangle (4)It may be either an acute or an obtuse triangle 3. If the lengths of the sides of a triangle are 3, 4, and 5 respectively, what is the cosine of the largest angle of the triangle?
3 3 4(1) 0 (2) (3) (4) 5 4 5
4. On a level piece of ground, two surveyors establish a baseline running from A to B, a distance of 120 meters. From A and from B, a point C is sighted and ∠CAB and ∠CBA are measured to be 51º 30’ and 93º 10’, respectively. Find AC to the nearest meter 5. From a ship, the angle of elevation of point A at the top of a cliff measures 22º. After the ship has sailed 2,000 feet directly toward the foot of the cliff, the angle of elevation of A measures 48º. Find, to the nearest ten feet, the height of the cliff.
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Trig Apps Classwork Day 8: Regents Practice - Review Warm-up: Regents Questions 1. A farmer has determined that a crop of strawberries yields a yearly profit of $1.50 per square yard. If
strawberries are planted on a triangular piece of land whose sides are 50 yards, 75 yards, and 100 yards, how much profit, to the nearest hundred dollars, would the farmer expect to make from this piece of land during the next harvest?
2. A triangular plot of land has sides that measure 5 meters, 7 meters, and 10 meters. What is the area of this
plot of land, to the nearest tenth of a square meter?
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3. Two forces act on a body, making angles of 15º and 37º
with the resultant. If the larger force is 50 pounds, what is the magnitude of the resultant to the nearest pound?
4. Gregory wants to build a garden in the shape of an
isosceles triangle with one of the congruent sides equal to 12 yards. If the area of his garden will be 55 square yards, find, to the nearest tenth of a degree, the three angles of the triangle.
5. In the accompanying diagram of ∆ABC, m∠A = 65,
m∠B = 70, and the side opposite vertex B is 7. Find the length of the side opposite vertex A, and find the area of ∆ABC.
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Trig Apps Homework Day 8 1. In ∆ABC, a = 20, b = 12, and m∠C = 44º a) Find the length of side c to the nearest integer. b) Find the area of ∆ABC to the nearest tenth. 2. In ∆ABC, m∠A = 50º, a = 40, b = 45, and ∠B is
obtuse. Find m∠C to the nearest degree. 3. How many distinct triangles can be constructed if
m∠A = 30º, a = 8, and b = 10? 4. From a ship, the angle of elevation of point A at the top of a
cliff measures 30º. After the ship has sailed 5,000ft directly toward the foot of the cliff, the angle of elevation of A measures 50º. Find to the nearest ten feet the height of the cliff.
5. A ship at sea heads directly toward a cliff on the
shoreline. The accompanying diagram shows the top of the cliff, D, sighted from two locations, A and B, separated by distance, S. If m∠DAC = 30, m∠DBC = 45, and S = 30 feet, what is the height of the cliff, to the nearest foot?
